English for Maths I



English for Maths I

Wk 6 Circles

TASK 1. Read the following description of circle properties and find the odd one out.

• The circle is the shape with the largest area for a given length of perimeter.

• The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group.

• All circles are similar:

o A circle's circumference and radius are proportional.

o The area enclosed and the square of its radius are disproportional.

o The constants of proportionality are 2π and π, respectively.

• The circle which is centred at the origin with radius 1 is called the unit circle.

o Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

• Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points.

Chord

TASK 2. Complete the missing information from the following text on Chord properties:

• Chords are equidistant from the centre of a circe ………………they are equal in length.

• The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:

o A perpendicular line from the centre of a circle bisects the chord.

o The line segment (circular segment) through the centre bisecting a chord is …………………….the chord.

• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is ………………….the inscribed angle.

• If two angles are inscribed on the same chord and on the same side of the chord, then they are…………………….

• If two angles are inscribed on the same chord and on opposite sides of the chord, then they are……………………..

o For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

• The ………………….is the longest chord of the circle.

• An inscribed angle …………by a diameter is a right angle (Thales' theorem).

• If the …………………….of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.

• If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the ………………………….

• The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).

• The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.

Sagitta

TASK 3. Read the short description of Sagitta and choose the correct option.

[pic]

The sagitta is the perpendicular/vertical/horizontal segment.

• The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the centre/midpoint/endpoint of that chord and the arc of the circle.

• Given/Provided/Taking into account the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines:

[pic]

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta subtends/divides/bisects/intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find/come to the conclusion/discover that (2r − x)x = (y / 2)2. Solving for r, we find the required result.

Tangent Properties

TASK 4. How many Tangent properties hold TRUE in the following picture?

• A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.

• A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.

• Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

• If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.

• If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ).

Secant-secant theorem

TASK 5. Complete the missing words.

• The chord theorem states that if two chords, CD and EB, intersect ……. A, then CA × DA = EA × BA.

• If a tangent from an external point D meets the circle at C and a secant ……. the external point D meets the circle at G and E respectively, then DC2 = DG × DE. (Tangent-secant theorem.)

• If two secants, DG and DE, also cut the circle at H and F ……………., then DH × DG = DF × DE.

• The angle between a tangent and chord is equal ………. one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).

• If the angle subtended by the chord …….. the centre is 90 degrees then l = r √2, where l is the length of the chord and r is the radius of the circle.

• If two secants are inscribed in the circle as shown …………. right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC).

Inscribed angles

TASK 6. Read the following text on Inscribed angles and write the missing numbers that correspond to the examples.

Inscribed angle theorem

An inscribed angle (examples in the figure are …?) is exactly half the corresponding central angle (…?). Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

TASK 7. Group Work:

Chord properties

In groups of three come up with a solution for the following problems

[pic]

Chord and tangent properties

In groups of three come up with a solution for the following problems:

[pic]

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